Random matrix theory of open quantum systems

Describing interacting quantum systems in the presence of dissipation is a daunting task. One very fruitful approach to this problem, developed for unitary dynamics, is to represent the Hamiltonian of a generic system by a large random matrix. This approach eventually led to the development of the field of quantum chaos.

Here, I will give an overview of the recent application of (non-Hermitian) random matrix theory to open quantum systems, where dissipation and decoherence coexist with unitary dynamics. I will discuss the spectral features, relaxation timescales, and steady states of three representative examples of increasing physical relevance: fully random Lindbladians and quantum maps, random open free fermions, and dissipative stronglyinteracting SYK models.

Then, I will argue that complex spacing ratios are a powerful signature of dissipative quantum chaos, illustrating it with the case of boundary-driven spin-chain Lindbladians.